= Solution
\b[Yes: the same physical resonant cavity and its field remain resonant in every inertial description.] A <Lorentz transformation> changes the mirror worldlines and the field together. The boundary <phase matching> and the constructive-interference condition are statements about the same reflection events, and the contracted <wave phase> is a <Lorentz scalar>. A boosted resonant solution therefore still satisfies the moving-mirror <boundary conditions>.
It would be incorrect merely to insert a Lorentz-contracted length into the formula for two stationary mirrors while leaving both propagation <frequencies> unchanged. For example, let the mirrors be separated by $L$ in their <rest frame> and let $\omega_0L/c=m\pi$, including the usual equal-mirror reflection-phase convention. For an observer boosted at $+v$ along the cavity axis, the cavity moves at $-v$ and its two travelling components have <frequencies>
$$
\omega'_+=\gamma\omega_0(1-\beta),\qquad
\omega'_- =\gamma\omega_0(1+\beta).
$$
They need not have one common laboratory <frequency>. The simultaneous mirror separation is $L'=L/\gamma$, while the forward and return flight times are $L'/(c+v)$ and $L'/(c-v)$. The round-trip interval between events on the same mirror is therefore $2\gamma L/c$.
Along that moving mirror the <wave phase> rate of either wave component is $\omega'-k'v_{\rm mirror}=\omega_0/\gamma$. Its advance over the round trip is
$$
\boxed{\frac{\omega_0}{\gamma}\frac{2\gamma L}{c}=\frac{2\omega_0L}{c}=2\pi m.}
$$
Thus the <resonance> condition is unchanged even though the observed travelling <frequencies>, lengths and flight times all change. For a boost in a general direction the beam directions also undergo <relativistic aberration>; the event-by-event phase-invariance argument still applies. A separate stationary cavity placed in the new frame, or an external laser whose <frequency> is not transformed with the original setup, is a different physical system.
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